LMFDB - Embedded newform 1764.4.f.b.881.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(881,1764)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1764.881");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$104.079369250$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.15
Character	$\chi$	$=$	1764.881
Dual form			1764.4.f.b.881.16

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.55832 q^{5} +O(q^{10})$	$q+6.55832 q^{5} +39.5400i q^{11} -79.8324i q^{13} +68.3138 q^{17} +11.4204i q^{19} +136.258i q^{23} -81.9884 q^{25} +9.32277i q^{29} +80.8126i q^{31} +235.271 q^{37} +50.7669 q^{41} -24.3676 q^{43} +228.958 q^{47} -477.559i q^{53} +259.316i q^{55} -389.183 q^{59} -398.839i q^{61} -523.566i q^{65} +754.874 q^{67} +799.714i q^{71} +604.177i q^{73} +45.8387 q^{79} +45.6855 q^{83} +448.024 q^{85} +326.627 q^{89} +74.8985i q^{95} -109.900i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 888 q^{25} + 864 q^{37} - 1248 q^{43} + 1056 q^{67} - 8064 q^{85}+O(q^{100}) $ 24 * q + 888 * q^25 + 864 * q^37 - 1248 * q^43 + 1056 * q^67 - 8064 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	6.55832	0.586594	0.293297	−	0.956021i	$-0.405247\pi$
$5$	6.55832	0.586594	0.293297	+	0.956021i	$0.405247\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	39.5400i	1.08380i	0.840444	+	0.541899i	$0.182294\pi$
$11$	39.5400i	1.08380i	−0.840444	+	0.541899i	$0.817706\pi$
$12$	0	0
$13$	− 79.8324i	− 1.70319i	−0.524197	−	0.851597i	$-0.675635\pi$
$13$	− 79.8324i	− 1.70319i	0.524197	−	0.851597i	$-0.324365\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	68.3138	0.974620	0.487310	−	0.873229i	$-0.337978\pi$
$17$	68.3138	0.974620	0.487310	+	0.873229i	$0.337978\pi$
$18$	0	0
$19$	11.4204i	0.137896i	0.997620	+	0.0689478i	$0.0219642\pi$
$19$	11.4204i	0.137896i	−0.997620	+	0.0689478i	$0.978036\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	136.258i	1.23529i	0.786456	+	0.617646i	$0.211914\pi$
$23$	136.258i	1.23529i	−0.786456	+	0.617646i	$0.788086\pi$
$24$	0	0
$25$	−81.9884	−0.655907
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.32277i	0.0596964i	0.999554	+	0.0298482i	$0.00950239\pi$
$29$	9.32277i	0.0596964i	−0.999554	+	0.0298482i	$0.990498\pi$
$30$	0	0
$31$	80.8126i	0.468205i	0.972212	+	0.234103i	$0.0752152\pi$
$31$	80.8126i	0.468205i	−0.972212	+	0.234103i	$0.924785\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	235.271	1.04536	0.522681	−	0.852529i	$-0.324932\pi$
$37$	235.271	1.04536	0.522681	+	0.852529i	$0.324932\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	50.7669	0.193377	0.0966886	−	0.995315i	$-0.469175\pi$
$41$	50.7669	0.193377	0.0966886	+	0.995315i	$0.469175\pi$
$42$	0	0
$43$	−24.3676	−0.0864191	−0.0432096	−	0.999066i	$-0.513758\pi$
$43$	−24.3676	−0.0864191	−0.0432096	+	0.999066i	$0.513758\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	228.958	0.710574	0.355287	−	0.934757i	$-0.384383\pi$
$47$	228.958	0.710574	0.355287	+	0.934757i	$0.384383\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 477.559i	− 1.23769i	−0.785511	−	0.618847i	$-0.787600\pi$
$53$	− 477.559i	− 1.23769i	0.785511	−	0.618847i	$-0.212400\pi$
$54$	0	0
$55$	259.316i	0.635749i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−389.183	−0.858767	−0.429384	−	0.903122i	$-0.641269\pi$
$59$	−389.183	−0.858767	−0.429384	+	0.903122i	$0.641269\pi$
$60$	0	0
$61$	− 398.839i	− 0.837149i	−0.908183	−	0.418574i	$-0.862530\pi$
$61$	− 398.839i	− 0.837149i	0.908183	−	0.418574i	$-0.137470\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 523.566i	− 0.999084i
$66$	0	0
$67$	754.874	1.37646	0.688228	−	0.725494i	$-0.258389\pi$
$67$	754.874	1.37646	0.688228	+	0.725494i	$0.258389\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	799.714i	1.33674i	0.743829	+	0.668370i	$0.233008\pi$
$71$	799.714i	1.33674i	−0.743829	+	0.668370i	$0.766992\pi$
$72$	0	0
$73$	604.177i	0.968679i	0.874880	+	0.484339i	$0.160940\pi$
$73$	604.177i	0.968679i	−0.874880	+	0.484339i	$0.839060\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	45.8387	0.0652817	0.0326408	−	0.999467i	$-0.489608\pi$
$79$	45.8387	0.0652817	0.0326408	+	0.999467i	$0.489608\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	45.6855	0.0604172	0.0302086	−	0.999544i	$-0.490383\pi$
$83$	45.6855	0.0604172	0.0302086	+	0.999544i	$0.490383\pi$
$84$	0	0
$85$	448.024	0.571706
$86$	0	0
$87$	0	0
$88$	0	0
$89$	326.627	0.389015	0.194508	−	0.980901i	$-0.437689\pi$
$89$	326.627	0.389015	0.194508	+	0.980901i	$0.437689\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	74.8985i	0.0808887i
$96$	0	0
$97$	− 109.900i	− 0.115038i	−0.998344	−	0.0575191i	$-0.981681\pi$
$97$	− 109.900i	− 0.115038i	0.998344	−	0.0575191i	$-0.0183190\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1977.41	1.94812	0.974059	−	0.226293i	$-0.0726607\pi$
$101$	1977.41	1.94812	0.974059	+	0.226293i	$0.0726607\pi$
$102$	0	0
$103$	1754.01i	1.67794i	0.544177	+	0.838971i	$0.316842\pi$
$103$	1754.01i	1.67794i	−0.544177	+	0.838971i	$0.683158\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	725.555i	0.655533i	0.944759	+	0.327767i	$0.106296\pi$
$107$	725.555i	0.655533i	−0.944759	+	0.327767i	$0.893704\pi$
$108$	0	0
$109$	−594.996	−0.522846	−0.261423	−	0.965224i	$-0.584192\pi$
$109$	−594.996	−0.522846	−0.261423	+	0.965224i	$0.584192\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	311.261i	0.259123i	0.991571	+	0.129562i	$0.0413570\pi$
$113$	311.261i	0.259123i	−0.991571	+	0.129562i	$0.958643\pi$
$114$	0	0
$115$	893.623i	0.724615i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−232.414	−0.174616
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1357.50	−0.971346
$126$	0	0
$127$	−2397.08	−1.67485	−0.837426	−	0.546551i	$-0.815940\pi$
$127$	−2397.08	−1.67485	−0.837426	+	0.546551i	$0.815940\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1334.18	0.889832	0.444916	−	0.895572i	$-0.353233\pi$
$131$	1334.18	0.889832	0.444916	+	0.895572i	$0.353233\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1135.62i	− 0.708193i	−0.935209	−	0.354097i	$-0.884788\pi$
$137$	− 1135.62i	− 0.708193i	0.935209	−	0.354097i	$-0.115212\pi$
$138$	0	0
$139$	− 1869.35i	− 1.14069i	−0.821404	−	0.570347i	$-0.806809\pi$
$139$	− 1869.35i	− 1.14069i	0.821404	−	0.570347i	$-0.193191\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3156.58	1.84592
$144$	0	0
$145$	61.1418i	0.0350176i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 977.391i	− 0.537389i	−0.963225	−	0.268694i	$-0.913408\pi$
$149$	− 977.391i	− 0.537389i	0.963225	−	0.268694i	$-0.0865922\pi$
$150$	0	0
$151$	1034.92	0.557751	0.278876	−	0.960327i	$-0.410038\pi$
$151$	1034.92	0.557751	0.278876	+	0.960327i	$0.410038\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	529.995i	0.274647i
$156$	0	0
$157$	62.3817i	0.0317108i	0.999874	+	0.0158554i	$0.00504715\pi$
$157$	62.3817i	0.0317108i	−0.999874	+	0.0158554i	$0.994953\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	397.781	0.191145	0.0955725	−	0.995422i	$-0.469532\pi$
$163$	397.781	0.191145	0.0955725	+	0.995422i	$0.469532\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2001.69	0.927519	0.463759	−	0.885961i	$-0.346500\pi$
$167$	2001.69	0.927519	0.463759	+	0.885961i	$0.346500\pi$
$168$	0	0
$169$	−4176.21	−1.90087
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1999.88	0.878892	0.439446	−	0.898269i	$-0.355175\pi$
$173$	1999.88	0.878892	0.439446	+	0.898269i	$0.355175\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3798.12i	− 1.58595i	−0.609255	−	0.792975i	$-0.708531\pi$
$179$	− 3798.12i	− 1.58595i	0.609255	−	0.792975i	$-0.291469\pi$
$180$	0	0
$181$	2540.89i	1.04344i	0.853116	+	0.521720i	$0.174710\pi$
$181$	2540.89i	1.04344i	−0.853116	+	0.521720i	$0.825290\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1542.99	0.613203
$186$	0	0
$187$	2701.13i	1.05629i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 5235.99i	− 1.98358i	−0.127889	−	0.991789i	$-0.540820\pi$
$191$	− 5235.99i	− 1.98358i	0.127889	−	0.991789i	$-0.459180\pi$
$192$	0	0
$193$	5115.18	1.90777	0.953883	−	0.300178i	$-0.0970461\pi$
$193$	5115.18	1.90777	0.953883	+	0.300178i	$0.0970461\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3849.94i	1.39237i	0.717862	+	0.696186i	$0.245121\pi$
$197$	3849.94i	1.39237i	−0.717862	+	0.696186i	$0.754879\pi$
$198$	0	0
$199$	2430.44i	0.865774i	0.901448	+	0.432887i	$0.142505\pi$
$199$	2430.44i	0.865774i	−0.901448	+	0.432887i	$0.857495\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	332.946	0.113434
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−451.562	−0.149451
$210$	0	0
$211$	4284.81	1.39800	0.699000	−	0.715121i	$-0.253629\pi$
$211$	4284.81	1.39800	0.699000	+	0.715121i	$0.253629\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−159.811	−0.0506930
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 5453.66i	− 1.65997i
$222$	0	0
$223$	1484.34i	0.445735i	0.974849	+	0.222868i	$0.0715418\pi$
$223$	1484.34i	0.445735i	−0.974849	+	0.222868i	$0.928458\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4711.69	1.37765	0.688823	−	0.724930i	$-0.258128\pi$
$227$	4711.69	1.37765	0.688823	+	0.724930i	$0.258128\pi$
$228$	0	0
$229$	3906.10i	1.12717i	0.826057	+	0.563586i	$0.190579\pi$
$229$	3906.10i	1.12717i	−0.826057	+	0.563586i	$0.809421\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2782.31i	0.782297i	0.920328	+	0.391148i	$0.127922\pi$
$233$	2782.31i	0.782297i	−0.920328	+	0.391148i	$0.872078\pi$
$234$	0	0
$235$	1501.58	0.416819
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6237.16i	1.68807i	0.536289	+	0.844034i	$0.319826\pi$
$239$	6237.16i	1.68807i	−0.536289	+	0.844034i	$0.680174\pi$
$240$	0	0
$241$	5892.06i	1.57486i	0.616405	+	0.787429i	$0.288589\pi$
$241$	5892.06i	1.57486i	−0.616405	+	0.787429i	$0.711411\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	911.716	0.234863
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4252.57	−1.06940	−0.534701	−	0.845041i	$-0.679576\pi$
$251$	−4252.57	−1.06940	−0.534701	+	0.845041i	$0.679576\pi$
$252$	0	0
$253$	−5387.64	−1.33881
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6377.98	1.54804	0.774022	−	0.633158i	$-0.218242\pi$
$257$	6377.98	1.54804	0.774022	+	0.633158i	$0.218242\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3012.56i	0.706321i	0.935563	+	0.353160i	$0.114893\pi$
$263$	3012.56i	0.706321i	−0.935563	+	0.353160i	$0.885107\pi$
$264$	0	0
$265$	− 3131.99i	− 0.726024i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4785.30	1.08463	0.542314	−	0.840176i	$-0.317548\pi$
$269$	4785.30	1.08463	0.542314	+	0.840176i	$0.317548\pi$
$270$	0	0
$271$	− 1016.29i	− 0.227805i	−0.993492	−	0.113902i	$-0.963665\pi$
$271$	− 1016.29i	− 0.227805i	0.993492	−	0.113902i	$-0.0363351\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3241.82i	− 0.710870i
$276$	0	0
$277$	−668.924	−0.145097	−0.0725483	−	0.997365i	$-0.523113\pi$
$277$	−668.924	−0.145097	−0.0725483	+	0.997365i	$0.523113\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2593.65i	0.550620i	0.961355	+	0.275310i	$0.0887805\pi$
$281$	2593.65i	0.550620i	−0.961355	+	0.275310i	$0.911219\pi$
$282$	0	0
$283$	6344.60i	1.33268i	0.745650	+	0.666338i	$0.232139\pi$
$283$	6344.60i	1.33268i	−0.745650	+	0.666338i	$0.767861\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−246.222	−0.0501164
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1039.64	−0.207292	−0.103646	−	0.994614i	$-0.533051\pi$
$293$	−1039.64	−0.207292	−0.103646	+	0.994614i	$0.533051\pi$
$294$	0	0
$295$	−2552.39	−0.503748
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10877.8	2.10394
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2615.71i	− 0.491066i
$306$	0	0
$307$	− 3398.31i	− 0.631765i	−0.948798	−	0.315882i	$-0.897699\pi$
$307$	− 3398.31i	− 0.631765i	0.948798	−	0.315882i	$-0.102301\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9513.78	1.73465	0.867326	−	0.497740i	$-0.165837\pi$
$311$	9513.78	1.73465	0.867326	+	0.497740i	$0.165837\pi$
$312$	0	0
$313$	− 950.996i	− 0.171736i	−0.996307	−	0.0858682i	$-0.972634\pi$
$313$	− 950.996i	− 0.171736i	0.996307	−	0.0858682i	$-0.0273664\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7216.78i	1.27866i	0.768933	+	0.639329i	$0.220788\pi$
$317$	7216.78i	1.27866i	−0.768933	+	0.639329i	$0.779212\pi$
$318$	0	0
$319$	−368.623	−0.0646988
$320$	0	0
$321$	0	0
$322$	0	0
$323$	780.170i	0.134396i
$324$	0	0
$325$	6545.33i	1.11714i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8731.06	1.44986	0.724928	−	0.688824i	$-0.241873\pi$
$331$	8731.06	1.44986	0.724928	+	0.688824i	$0.241873\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4950.71	0.807421
$336$	0	0
$337$	−5333.20	−0.862071	−0.431035	−	0.902335i	$-0.641852\pi$
$337$	−5333.20	−0.862071	−0.431035	+	0.902335i	$0.641852\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3195.33	−0.507440
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 10591.3i	− 1.63854i	−0.573408	−	0.819270i	$-0.694379\pi$
$347$	− 10591.3i	− 1.63854i	0.573408	−	0.819270i	$-0.305621\pi$
$348$	0	0
$349$	− 4590.04i	− 0.704009i	−0.935998	−	0.352005i	$-0.885500\pi$
$349$	− 4590.04i	− 0.704009i	0.935998	−	0.352005i	$-0.114500\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1558.06	0.234922	0.117461	−	0.993078i	$-0.462525\pi$
$353$	1558.06	0.234922	0.117461	+	0.993078i	$0.462525\pi$
$354$	0	0
$355$	5244.78i	0.784124i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 158.688i	− 0.0233293i	−0.999932	−	0.0116647i	$-0.996287\pi$
$359$	− 158.688i	− 0.0233293i	0.999932	−	0.0116647i	$-0.00371306\pi$
$360$	0	0
$361$	6728.57	0.980985
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3962.39i	0.568221i
$366$	0	0
$367$	− 6559.23i	− 0.932940i	−0.884537	−	0.466470i	$-0.845526\pi$
$367$	− 6559.23i	− 0.932940i	0.884537	−	0.466470i	$-0.154474\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4364.23	0.605821	0.302911	−	0.953019i	$-0.402042\pi$
$373$	4364.23	0.605821	0.302911	+	0.953019i	$0.402042\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	744.259	0.101675
$378$	0	0
$379$	2376.69	0.322117	0.161058	−	0.986945i	$-0.448509\pi$
$379$	2376.69	0.322117	0.161058	+	0.986945i	$0.448509\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1294.13	0.172656	0.0863279	−	0.996267i	$-0.472487\pi$
$383$	1294.13	0.172656	0.0863279	+	0.996267i	$0.472487\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 9600.23i	− 1.25129i	−0.780109	−	0.625644i	$-0.784836\pi$
$389$	− 9600.23i	− 1.25129i	0.780109	−	0.625644i	$-0.215164\pi$
$390$	0	0
$391$	9308.29i	1.20394i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	300.625	0.0382939
$396$	0	0
$397$	3272.83i	0.413750i	0.978367	+	0.206875i	$0.0663293\pi$
$397$	3272.83i	0.413750i	−0.978367	+	0.206875i	$0.933671\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 14617.9i	− 1.82041i	−0.414157	−	0.910205i	$-0.635924\pi$
$401$	− 14617.9i	− 1.82041i	0.414157	−	0.910205i	$-0.364076\pi$
$402$	0	0
$403$	6451.46	0.797445
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9302.64i	1.13296i
$408$	0	0
$409$	3017.63i	0.364822i	0.983222	+	0.182411i	$0.0583902\pi$
$409$	3017.63i	0.364822i	−0.983222	+	0.182411i	$0.941610\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	299.620	0.0354404
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7955.52	0.927572	0.463786	−	0.885947i	$-0.346491\pi$
$419$	7955.52	0.927572	0.463786	+	0.885947i	$0.346491\pi$
$420$	0	0
$421$	4238.67	0.490689	0.245344	−	0.969436i	$-0.421099\pi$
$421$	4238.67	0.490689	0.245344	+	0.969436i	$0.421099\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5600.94	−0.639260
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8911.41i	0.995934i	0.867196	+	0.497967i	$0.165920\pi$
$431$	8911.41i	0.995934i	−0.867196	+	0.497967i	$0.834080\pi$
$432$	0	0
$433$	− 427.546i	− 0.0474517i	−0.999719	−	0.0237258i	$-0.992447\pi$
$433$	− 427.546i	− 0.0474517i	0.999719	−	0.0237258i	$-0.00755287\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1556.12	−0.170341
$438$	0	0
$439$	5784.54i	0.628886i	0.949276	+	0.314443i	$0.101818\pi$
$439$	5784.54i	0.628886i	−0.949276	+	0.314443i	$0.898182\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	203.165i	0.0217893i	0.999941	+	0.0108947i	$0.00346795\pi$
$443$	203.165i	0.0217893i	−0.999941	+	0.0108947i	$0.996532\pi$
$444$	0	0
$445$	2142.12	0.228194
$446$	0	0
$447$	0	0
$448$	0	0
$449$	609.140i	0.0640247i	0.999487	+	0.0320124i	$0.0101916\pi$
$449$	609.140i	0.0640247i	−0.999487	+	0.0320124i	$0.989808\pi$
$450$	0	0
$451$	2007.33i	0.209582i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12270.6	−1.25600	−0.628001	−	0.778212i	$-0.716127\pi$
$457$	−12270.6	−1.25600	−0.628001	+	0.778212i	$0.716127\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15165.2	−1.53213	−0.766066	−	0.642762i	$-0.777788\pi$
$461$	−15165.2	−1.53213	−0.766066	+	0.642762i	$0.777788\pi$
$462$	0	0
$463$	−14315.9	−1.43697	−0.718485	−	0.695542i	$-0.755164\pi$
$463$	−14315.9	−1.43697	−0.718485	+	0.695542i	$0.755164\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16964.2	1.68097	0.840483	−	0.541837i	$-0.182271\pi$
$467$	16964.2	1.68097	0.840483	+	0.541837i	$0.182271\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 963.495i	− 0.0936608i
$474$	0	0
$475$	− 936.339i	− 0.0904467i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3599.84	−0.343384	−0.171692	−	0.985151i	$-0.554923\pi$
$479$	−3599.84	−0.343384	−0.171692	+	0.985151i	$0.554923\pi$
$480$	0	0
$481$	− 18782.3i	− 1.78045i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 720.763i	− 0.0674807i
$486$	0	0
$487$	−5504.57	−0.512189	−0.256094	−	0.966652i	$-0.582436\pi$
$487$	−5504.57	−0.512189	−0.256094	+	0.966652i	$0.582436\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 13643.7i	− 1.25404i	−0.779003	−	0.627020i	$-0.784274\pi$
$491$	− 13643.7i	− 1.25404i	0.779003	−	0.627020i	$-0.215726\pi$
$492$	0	0
$493$	636.874i	0.0581813i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−17072.6	−1.53161	−0.765807	−	0.643070i	$-0.777660\pi$
$499$	−17072.6	−1.53161	−0.765807	+	0.643070i	$0.777660\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7574.12	−0.671398	−0.335699	−	0.941969i	$-0.608972\pi$
$503$	−7574.12	−0.671398	−0.335699	+	0.941969i	$0.608972\pi$
$504$	0	0
$505$	12968.5	1.14275
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16920.2	−1.47343	−0.736713	−	0.676206i	$-0.763623\pi$
$509$	−16920.2	−1.47343	−0.736713	+	0.676206i	$0.763623\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11503.4i	0.984270i
$516$	0	0
$517$	9053.02i	0.770119i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21080.5	1.77265	0.886326	−	0.463061i	$-0.153249\pi$
$521$	21080.5	1.77265	0.886326	+	0.463061i	$0.153249\pi$
$522$	0	0
$523$	17723.2i	1.48180i	0.671613	+	0.740902i	$0.265602\pi$
$523$	17723.2i	1.48180i	−0.671613	+	0.740902i	$0.734398\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5520.62i	0.456322i
$528$	0	0
$529$	−6399.20	−0.525947
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4052.85i	− 0.329359i
$534$	0	0
$535$	4758.42i	0.384532i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−766.541	−0.0609171	−0.0304586	−	0.999536i	$-0.509697\pi$
$541$	−766.541	−0.0609171	−0.0304586	+	0.999536i	$0.509697\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3902.17	−0.306699
$546$	0	0
$547$	−6579.52	−0.514296	−0.257148	−	0.966372i	$-0.582783\pi$
$547$	−6579.52	−0.514296	−0.257148	+	0.966372i	$0.582783\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−106.470	−0.00823187
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9020.19i	0.686172i	0.939304	+	0.343086i	$0.111472\pi$
$557$	9020.19i	0.686172i	−0.939304	+	0.343086i	$0.888528\pi$
$558$	0	0
$559$	1945.32i	0.147189i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21088.7	−1.57866	−0.789328	−	0.613972i	$-0.789571\pi$
$563$	−21088.7	−1.57866	−0.789328	+	0.613972i	$0.789571\pi$
$564$	0	0
$565$	2041.35i	0.152000i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15803.4i	1.16435i	0.813064	+	0.582174i	$0.197798\pi$
$569$	15803.4i	1.16435i	−0.813064	+	0.582174i	$0.802202\pi$
$570$	0	0
$571$	−26183.7	−1.91901	−0.959505	−	0.281693i	$-0.909104\pi$
$571$	−26183.7	−1.91901	−0.959505	+	0.281693i	$0.909104\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 11171.6i	− 0.810237i
$576$	0	0
$577$	− 23546.5i	− 1.69888i	−0.527687	−	0.849439i	$-0.676941\pi$
$577$	− 23546.5i	− 1.69888i	0.527687	−	0.849439i	$-0.323059\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	18882.7	1.34141
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12296.8	0.864640	0.432320	−	0.901720i	$-0.357695\pi$
$587$	12296.8	0.864640	0.432320	+	0.901720i	$0.357695\pi$
$588$	0	0
$589$	−922.911	−0.0645634
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9743.96	−0.674767	−0.337383	−	0.941367i	$-0.609542\pi$
$593$	−9743.96	−0.674767	−0.337383	+	0.941367i	$0.609542\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4272.74i	0.291451i	0.989325	+	0.145726i	$0.0465517\pi$
$599$	4272.74i	0.291451i	−0.989325	+	0.145726i	$0.953448\pi$
$600$	0	0
$601$	− 28823.5i	− 1.95630i	−0.207903	−	0.978149i	$-0.566664\pi$
$601$	− 28823.5i	− 1.95630i	0.207903	−	0.978149i	$-0.433336\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1524.25	−0.102429
$606$	0	0
$607$	− 8090.23i	− 0.540976i	−0.962723	−	0.270488i	$-0.912815\pi$
$607$	− 8090.23i	− 0.540976i	0.962723	−	0.270488i	$-0.0871850\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 18278.3i	− 1.21025i
$612$	0	0
$613$	16997.4	1.11993	0.559966	−	0.828516i	$-0.310814\pi$
$613$	16997.4	1.11993	0.559966	+	0.828516i	$0.310814\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4408.19i	0.287629i	0.989605	+	0.143815i	$0.0459369\pi$
$617$	4408.19i	0.287629i	−0.989605	+	0.143815i	$0.954063\pi$
$618$	0	0
$619$	− 15207.3i	− 0.987454i	−0.869617	−	0.493727i	$-0.835634\pi$
$619$	− 15207.3i	− 0.987454i	0.869617	−	0.493727i	$-0.164366\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1345.65	0.0861217
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16072.3	1.01883
$630$	0	0
$631$	17710.9	1.11737	0.558685	−	0.829380i	$-0.311306\pi$
$631$	17710.9	1.11737	0.558685	+	0.829380i	$0.311306\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−15720.8	−0.982458
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 10639.4i	− 0.655585i	−0.944750	−	0.327793i	$-0.893695\pi$
$641$	− 10639.4i	− 0.655585i	0.944750	−	0.327793i	$-0.106305\pi$
$642$	0	0
$643$	14402.2i	0.883306i	0.897186	+	0.441653i	$0.145608\pi$
$643$	14402.2i	0.883306i	−0.897186	+	0.441653i	$0.854392\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27539.8	−1.67342	−0.836710	−	0.547646i	$-0.815524\pi$
$647$	−27539.8	−1.67342	−0.836710	+	0.547646i	$0.815524\pi$
$648$	0	0
$649$	− 15388.3i	− 0.930730i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15199.5i	0.910880i	0.890267	+	0.455440i	$0.150518\pi$
$653$	15199.5i	0.910880i	−0.890267	+	0.455440i	$0.849482\pi$
$654$	0	0
$655$	8750.00	0.521971
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 17308.4i	− 1.02313i	−0.859246	−	0.511563i	$-0.829067\pi$
$659$	− 17308.4i	− 1.02313i	0.859246	−	0.511563i	$-0.170933\pi$
$660$	0	0
$661$	− 18946.4i	− 1.11487i	−0.830220	−	0.557436i	$-0.811785\pi$
$661$	− 18946.4i	− 1.11487i	0.830220	−	0.557436i	$-0.188215\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1270.30	−0.0737425
$668$	0	0
$669$	0	0
$670$	0	0
$671$	15770.1	0.907299
$672$	0	0
$673$	992.154	0.0568273	0.0284136	−	0.999596i	$-0.490954\pi$
$673$	992.154	0.0568273	0.0284136	+	0.999596i	$0.490954\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−20381.5	−1.15705	−0.578526	−	0.815664i	$-0.696372\pi$
$677$	−20381.5	−1.15705	−0.578526	+	0.815664i	$0.696372\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16779.3i	0.940030i	0.882658	+	0.470015i	$0.155752\pi$
$683$	16779.3i	0.940030i	−0.882658	+	0.470015i	$0.844248\pi$
$684$	0	0
$685$	− 7447.75i	− 0.415422i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−38124.7	−2.10803
$690$	0	0
$691$	− 15106.5i	− 0.831664i	−0.909442	−	0.415832i	$-0.863491\pi$
$691$	− 15106.5i	− 0.831664i	0.909442	−	0.415832i	$-0.136509\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 12259.8i	− 0.669125i
$696$	0	0
$697$	3468.08	0.188469
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 4771.58i	− 0.257090i	−0.991704	−	0.128545i	$-0.958969\pi$
$701$	− 4771.58i	− 0.257090i	0.991704	−	0.128545i	$-0.0410306\pi$
$702$	0	0
$703$	2686.89i	0.144151i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6378.76	−0.337883	−0.168942	−	0.985626i	$-0.554035\pi$
$709$	−6378.76	−0.337883	−0.168942	+	0.985626i	$0.554035\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−11011.4	−0.578371
$714$	0	0
$715$	20701.8	1.08280
$716$	0	0
$717$	0	0
$718$	0	0
$719$	26840.0	1.39216	0.696079	−	0.717965i	$-0.254926\pi$
$719$	26840.0	1.39216	0.696079	+	0.717965i	$0.254926\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 764.359i	− 0.0391553i
$726$	0	0
$727$	13222.5i	0.674547i	0.941407	+	0.337273i	$0.109505\pi$
$727$	13222.5i	0.674547i	−0.941407	+	0.337273i	$0.890495\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1664.64	−0.0842258
$732$	0	0
$733$	− 21412.5i	− 1.07898i	−0.841994	−	0.539488i	$-0.818618\pi$
$733$	− 21412.5i	− 1.07898i	0.841994	−	0.539488i	$-0.181382\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	29847.8i	1.49180i
$738$	0	0
$739$	−34177.0	−1.70125	−0.850624	−	0.525774i	$-0.823776\pi$
$739$	−34177.0	−1.70125	−0.850624	+	0.525774i	$0.823776\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 23205.8i	− 1.14581i	−0.819621	−	0.572906i	$-0.805816\pi$
$743$	− 23205.8i	− 1.14581i	0.819621	−	0.572906i	$-0.194184\pi$
$744$	0	0
$745$	− 6410.04i	− 0.315229i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	18465.9	0.897246	0.448623	−	0.893721i	$-0.351915\pi$
$751$	18465.9	0.897246	0.448623	+	0.893721i	$0.351915\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6787.33	0.327174
$756$	0	0
$757$	13606.4	0.653280	0.326640	−	0.945149i	$-0.394084\pi$
$757$	13606.4	0.653280	0.326640	+	0.945149i	$0.394084\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18016.4	−0.858203	−0.429101	−	0.903256i	$-0.641170\pi$
$761$	−18016.4	−0.858203	−0.429101	+	0.903256i	$0.641170\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	31069.4i	1.46265i
$768$	0	0
$769$	21716.1i	1.01834i	0.860666	+	0.509169i	$0.170047\pi$
$769$	21716.1i	1.01834i	−0.860666	+	0.509169i	$0.829953\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−36210.0	−1.68484	−0.842420	−	0.538821i	$-0.818870\pi$
$773$	−36210.0	−1.68484	−0.842420	+	0.538821i	$0.818870\pi$
$774$	0	0
$775$	− 6625.70i	− 0.307099i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	579.778i	0.0266658i
$780$	0	0
$781$	−31620.7	−1.44876
$782$	0	0
$783$	0	0
$784$	0	0
$785$	409.119i	0.0186014i
$786$	0	0
$787$	15729.9i	0.712464i	0.934397	+	0.356232i	$0.115939\pi$
$787$	15729.9i	0.712464i	−0.934397	+	0.356232i	$0.884061\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−31840.2	−1.42583
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31098.5	−1.38214	−0.691070	−	0.722787i	$-0.742861\pi$
$797$	−31098.5	−1.38214	−0.691070	+	0.722787i	$0.742861\pi$
$798$	0	0
$799$	15641.0	0.692540
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−23889.2	−1.04985
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	25655.2i	1.11494i	0.830196	+	0.557471i	$0.188228\pi$
$809$	25655.2i	1.11494i	−0.830196	+	0.557471i	$0.811772\pi$
$810$	0	0
$811$	17351.0i	0.751263i	0.926769	+	0.375632i	$0.122574\pi$
$811$	17351.0i	0.751263i	−0.926769	+	0.375632i	$0.877426\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2608.78	0.112125
$816$	0	0
$817$	− 278.287i	− 0.0119168i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	23299.8i	0.990462i	0.868761	+	0.495231i	$0.164917\pi$
$821$	23299.8i	0.990462i	−0.868761	+	0.495231i	$0.835083\pi$
$822$	0	0
$823$	−14424.3	−0.610933	−0.305466	−	0.952203i	$-0.598812\pi$
$823$	−14424.3	−0.610933	−0.305466	+	0.952203i	$0.598812\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 11516.8i	− 0.484255i	−0.970244	−	0.242127i	$-0.922155\pi$
$827$	− 11516.8i	− 0.484255i	0.970244	−	0.242127i	$-0.0778452\pi$
$828$	0	0
$829$	9378.64i	0.392923i	0.980512	+	0.196462i	$0.0629451\pi$
$829$	9378.64i	0.392923i	−0.980512	+	0.196462i	$0.937055\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	13127.7	0.544077
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31050.7	−1.27770	−0.638848	−	0.769333i	$-0.720589\pi$
$839$	−31050.7	−1.27770	−0.638848	+	0.769333i	$0.720589\pi$
$840$	0	0
$841$	24302.1	0.996436
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−27388.9	−1.11504
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	32057.6i	1.29133i
$852$	0	0
$853$	− 41308.2i	− 1.65811i	−0.559170	−	0.829053i	$-0.688880\pi$
$853$	− 41308.2i	− 1.65811i	0.559170	−	0.829053i	$-0.311120\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27400.3	−1.09215	−0.546077	−	0.837735i	$-0.683879\pi$
$857$	−27400.3	−1.09215	−0.546077	+	0.837735i	$0.683879\pi$
$858$	0	0
$859$	3594.14i	0.142760i	0.997449	+	0.0713798i	$0.0227402\pi$
$859$	3594.14i	0.142760i	−0.997449	+	0.0713798i	$0.977260\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 3497.86i	− 0.137970i	−0.997618	−	0.0689852i	$-0.978024\pi$
$863$	− 3497.86i	− 0.137970i	0.997618	−	0.0689852i	$-0.0219761\pi$
$864$	0	0
$865$	13115.9	0.515553
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1812.46i	0.0707521i
$870$	0	0
$871$	− 60263.4i	− 2.34437i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−42495.4	−1.63622	−0.818111	−	0.575061i	$-0.804978\pi$
$877$	−42495.4	−1.63622	−0.818111	+	0.575061i	$0.804978\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5554.52	0.212414	0.106207	−	0.994344i	$-0.466129\pi$
$881$	5554.52	0.212414	0.106207	+	0.994344i	$0.466129\pi$
$882$	0	0
$883$	−1152.28	−0.0439154	−0.0219577	−	0.999759i	$-0.506990\pi$
$883$	−1152.28	−0.0439154	−0.0219577	+	0.999759i	$0.506990\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37737.6	1.42853	0.714264	−	0.699877i	$-0.246762\pi$
$887$	37737.6	1.42853	0.714264	+	0.699877i	$0.246762\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2614.79i	0.0979850i
$894$	0	0
$895$	− 24909.3i	− 0.930308i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−753.398	−0.0279502
$900$	0	0
$901$	− 32623.9i	− 1.20628i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16664.0i	0.612076i
$906$	0	0
$907$	−28129.6	−1.02980	−0.514900	−	0.857250i	$-0.672171\pi$
$907$	−28129.6	−1.02980	−0.514900	+	0.857250i	$0.672171\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 35517.2i	− 1.29170i	−0.763465	−	0.645849i	$-0.776503\pi$
$911$	− 35517.2i	− 1.29170i	0.763465	−	0.645849i	$-0.223497\pi$
$912$	0	0
$913$	1806.40i	0.0654800i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	19088.3	0.685162	0.342581	−	0.939488i	$-0.388699\pi$
$919$	19088.3	0.685162	0.342581	+	0.939488i	$0.388699\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	63843.1	2.27673
$924$	0	0
$925$	−19289.5	−0.685660
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−45972.1	−1.62357	−0.811784	−	0.583958i	$-0.801503\pi$
$929$	−45972.1	−1.62357	−0.811784	+	0.583958i	$0.801503\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	17714.9i	0.619614i
$936$	0	0
$937$	10653.2i	0.371423i	0.982604	+	0.185712i	$0.0594591\pi$
$937$	10653.2i	0.371423i	−0.982604	+	0.185712i	$0.940541\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6961.86	−0.241180	−0.120590	−	0.992702i	$-0.538479\pi$
$941$	−6961.86	−0.241180	−0.120590	+	0.992702i	$0.538479\pi$
$942$	0	0
$943$	6917.39i	0.238877i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 32401.9i	− 1.11185i	−0.831233	−	0.555924i	$-0.812365\pi$
$947$	− 32401.9i	− 1.11185i	0.831233	−	0.555924i	$-0.187635\pi$
$948$	0	0
$949$	48232.9	1.64985
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 27432.4i	− 0.932448i	−0.884667	−	0.466224i	$-0.845614\pi$
$953$	− 27432.4i	− 0.932448i	0.884667	−	0.466224i	$-0.154386\pi$
$954$	0	0
$955$	− 34339.3i	− 1.16355i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	23260.3	0.780784
$962$	0	0
$963$	0	0
$964$	0	0
$965$	33547.0	1.11908
$966$	0	0
$967$	43449.7	1.44493	0.722465	−	0.691408i	$-0.243009\pi$
$967$	43449.7	1.44493	0.722465	+	0.691408i	$0.243009\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−10546.6	−0.348565	−0.174283	−	0.984696i	$-0.555761\pi$
$971$	−10546.6	−0.348565	−0.174283	+	0.984696i	$0.555761\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 35389.5i	− 1.15887i	−0.815020	−	0.579433i	$-0.803274\pi$
$977$	− 35389.5i	− 1.15887i	0.815020	−	0.579433i	$-0.196726\pi$
$978$	0	0
$979$	12914.8i	0.421614i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27980.9	−0.907885	−0.453942	−	0.891031i	$-0.649983\pi$
$983$	−27980.9	−0.907885	−0.453942	+	0.891031i	$0.649983\pi$
$984$	0	0
$985$	25249.2i	0.816757i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 3320.28i	− 0.106753i
$990$	0	0
$991$	−10577.6	−0.339060	−0.169530	−	0.985525i	$-0.554225\pi$
$991$	−10577.6	−0.339060	−0.169530	+	0.985525i	$0.554225\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15939.6i	0.507858i
$996$	0	0
$997$	− 27653.2i	− 0.878420i	−0.898385	−	0.439210i	$-0.855258\pi$
$997$	− 27653.2i	− 0.878420i	0.898385	−	0.439210i	$-0.144742\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.f.b.881.15	yes	24
3.2	odd	2	inner	1764.4.f.b.881.10	yes	24
7.2	even	3		1764.4.t.c.521.10		48
7.3	odd	6		1764.4.t.c.1097.15		48
7.4	even	3		1764.4.t.c.1097.9		48
7.5	odd	6		1764.4.t.c.521.16		48
7.6	odd	2	inner	1764.4.f.b.881.9	✓	24
21.2	odd	6		1764.4.t.c.521.15		48
21.5	even	6		1764.4.t.c.521.9		48
21.11	odd	6		1764.4.t.c.1097.16		48
21.17	even	6		1764.4.t.c.1097.10		48
21.20	even	2	inner	1764.4.f.b.881.16	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.4.f.b.881.9	✓	24	7.6	odd	2	inner
1764.4.f.b.881.10	yes	24	3.2	odd	2	inner
1764.4.f.b.881.15	yes	24	1.1	even	1	trivial
1764.4.f.b.881.16	yes	24	21.20	even	2	inner
1764.4.t.c.521.9		48	21.5	even	6
1764.4.t.c.521.10		48	7.2	even	3
1764.4.t.c.521.15		48	21.2	odd	6
1764.4.t.c.521.16		48	7.5	odd	6
1764.4.t.c.1097.9		48	7.4	even	3
1764.4.t.c.1097.10		48	21.17	even	6
1764.4.t.c.1097.15		48	7.3	odd	6
1764.4.t.c.1097.16		48	21.11	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.55832 q^{5} +O(q^{10})\)	\(q+6.55832 q^{5} +39.5400i q^{11} -79.8324i q^{13} +68.3138 q^{17} +11.4204i q^{19} +136.258i q^{23} -81.9884 q^{25} +9.32277i q^{29} +80.8126i q^{31} +235.271 q^{37} +50.7669 q^{41} -24.3676 q^{43} +228.958 q^{47} -477.559i q^{53} +259.316i q^{55} -389.183 q^{59} -398.839i q^{61} -523.566i q^{65} +754.874 q^{67} +799.714i q^{71} +604.177i q^{73} +45.8387 q^{79} +45.6855 q^{83} +448.024 q^{85} +326.627 q^{89} +74.8985i q^{95} -109.900i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 888 q^{25} + 864 q^{37} - 1248 q^{43} + 1056 q^{67} - 8064 q^{85}+O(q^{100}) \) 24 * q + 888 * q^25 + 864 * q^37 - 1248 * q^43 + 1056 * q^67 - 8064 * q^85

Introduction

L-functions

Modular forms

Varieties

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Database

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